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This paper describes how to propagate approximately additive random perturbations through any kind of vision algorithm step in which the appropriate random perturbation model for the estimated quantity produced by the vision step is also an additive random perturbation. We assume that the vision algorithm step can be modeled as a calculation (linear or non-linear) that produces an estimate that minimizes an implicit scaler function of the input quantity and the calculated estimate. The only assumption is that the scaler function has finite second partial derivatives and that the random perturbations are small enough so that the relationship between the scaler function evaluated at the ideal but unknown input and output quantities and the observed input quantity and perturbed output quantity can be approximated sufficiently well by a first order Taylor series expansion.

The paper finally discusses the issues of verifying that the derived statistical behavior agrees with the experimentally observed statistical behavior.

In this paper, we study state-feedback controller design for controlling the Lyapunov exponents of an n-dimensional dynamical system. We examine some theoretical results and perform numerical simulations for systems with and without noise influence. The controlled Lyapunov exponents are asymptotically normally distributed if the system has noisy inputs. Computer simulations on finite samples are all consistent with the theoretical results.

Insects such as locusts and some moths can transform from a solitarious phase when they remain in loose populations and a gregarious phase, when they may swarm. Therefore, the key to effective management of outbreaks of species such as the desert locust Schistocercagregaria is early detection of when they are in the threshold state between the two phases, followed by timely control of their hopper stages before they fledge because the control of flying adult swarms is costly and often ineffective. Definitions of gregarization thresholds should assist preventive control measures and avoid treatment of areas that might not lead to gregarization. In order to better understand the effects of the threshold density which represents the gregarization threshold on the outbreak of a locust population, we developed a model of a discrete switching system. The proposed model allows us to address: (1) How frequently switching occurs from solitarious to gregarious phases and vice versa; (2) When do stable switching transients occur, the existence of which indicate that solutions with larger amplitudes can switch to a stable attractor with a value less than the switching threshold density?; and (3) How does random perturbation influence the switching pattern? Our results show that both subsystems have refuge equilibrium points, outbreak equilibrium points and bistable equilibria. Further, the outbreak equilibrium points and bistable equilibria can coexist for a wide range of parameters and can switch from one to another. This type of switching is sensitive to the intrinsic growth rate and the initial values of the locust population, and may result in locust population outbreaks and phase switching once a small perturbation occurs. Moreover, the simulation results indicate that the switching transient patterns become identical after some generations, suggesting that the evolving process of the perturbation system is not related to the initial value after some fixed number of generations for the same stochastic processes. However, the switching frequency and outbreak patterns can be significantly affected by the intensity of noise and the intrinsic growth rate of the locust population.

In ecosystems, almost all organisms suffer from external influences, which are assumed to be random perturbations. The aim of this paper is to investigate the effects of random perturbations on interacting populations. To solve this problem, we propose a discrete-time plant quality and larch budmoth interaction model with biological control. First of all, the model is theoretically analyzed, and the dynamic behavior of the population is qualitatively analyzed by numerical simulation. In addition, random perturbations are added to the model to observe the effects of random perturbations on the dynamic behavior of the budmoth population. The main results show that a random perturbation changes the number of blurred orbits of the system compared to the single population model with the same perturbation. Undeniably, it enhances the stability of the system. The model results, after adding random perturbations, are closer to reality, and in practice, we can also use random perturbations for resource management and pest control.

We prove that the statistical properties of random perturbations of a diffeomorphism with dominated splitting having mostly contracting center-stable direction and non-uniformly expanding center-unstable direction are described by a finite number of stationary measures. We also give necessary and sufficient conditions for the stochastic stability of such dynamical systems. We show that a certain C²-open class of non-uniformly hyperbolic diffeomorphisms introduced by Alves, Bonatti and Viana in [2] are stochastically stable.

The influence of small random perturbations on a deterministic dynamical system with a locally stable equilibrium is considered. The perturbed system is described by the Itô stochastic differential equation. It is assumed that the noise does not vanish at the equilibrium. In this case the trajectories of the stochastic system may leave any bounded domain with probability one. We analyze the stochastic stability of the equilibrium under a persistent perturbation by white noise on an asymptotically long time interval $0\le t\le {𝒪}({\mu }^{-N})$, where $0<\mu \ll 1$ is a perturbation parameter.

In this work, a fourth-order numerical scheme in space and two second-order numerical schemes in both time and space are proposed for the derivative nonlinear Schrödinger equation. We verify the mass conservation for the two-level implicit scheme. The influence on the soliton solution by adding a small random perturbation to the initial condition is discussed. The numerical experiments are given to test the accuracy order for different schemes, respectively. We also test the conservative property of mass and Hamiltonian for these schemes from the numerical point of view.

This paper describes how to propagate approximately additive random perturbations through any kind of vision algorithm step in which the appropriate random perturbation model for the estimated quantity produced by the vision step is also an additive random perturbation. We assume that the vision algorithm step can be modeled as a calculation (linear or non-linear) that produces an estimate that minimizes an implicit scaler function of the input quantity and the calculated estimate. The only assumption is that the scaler function has finite second partial derivatives and that the random perturbations are small enough so that the relationship between the scaler function evaluated at the ideal but unknown input and output quantities and the observed input quantity and perturbed output quantity can be approximated sufficiently well by a first order Taylor series expansion.

The paper finally discusses the issues of verifying that the derived statistical behavior agrees with the experimentally observed statistical behavior.

We consider KdV equation under periodic boundary conditions, perturbed by viscosity of size ν and a random force of size . For ν → 0 we derive averaged equations which describe behaviour of solutions when 0 ≤ t ≲ ν^-1 and when t → ∞.